Past to the left, the now line at the right. The pale band is the 256 plants, the thin line is the single best guess.
In plain words
Picture two water tanks stacked up, with a pump filling the top one. You want the bottom tank kept at a level you choose. The trouble is you only have a short, rough recording of the real tank, so you cannot know exactly how it behaves. The pale band on the chart is 256 slightly different guesses of the real tank, all at once. When the band is wide, nobody is sure what the water will do next.
Here is the trick. If you run the pump purely by your best guess and hope, the guesses drift apart and you lose track of the water. That is open loop. But if you keep watching the real level and nudge the pump every moment to correct it, the way a thermostat holds a room or cruise control holds a speed, all 256 guesses snap onto the target no matter which one is right. The band collapses to a single line. That is closed loop.
So the whole point is this: even when you cannot predict something well, steady correction lets you command it anyway. Press the big button to switch between guessing and correcting, and watch the band open and close.
What is honest here
The model is a believable twin, not an accurate one. Fit to the real record, its free-running test error is 0.593 V, essentially the same as a plain linear model at 0.588 V. It is good enough to command, not good enough to boast about.
The pale band is family spread. It is everything this five-parameter physical model can imagine given the short, noisy data, not literally every future the data cannot rule out. A real dataset is always poorer than the honest ideal, and this says so.
We expected the band to balloon at the overflow, the hard nonlinearity. It does not. An overflow is a clamp, and a clamp is contractive: it absorbs disagreeing futures rather than breeding them. Variance is bred by divergent dynamics, and two stable tanks and a ceiling have none.
What is left is the real lesson. Open the loop and the 256 plants disagree by about 0.16 V. Close the loop and they collapse to 0.006 V, roughly twenty-five times tighter, because integral feedback lands every stable plant on the target whatever its parameters. Feedback commands what the data cannot predict. That collapse is the whole piece.